propositionroutine

Complex network edges factor into translation kernels

Complex network edges factor into sc<Wcms_{\rm c}<W_{\rm c}m translation kernels
gives the kernel factorization the layer construction uses
Proposition·prop:complex-motif-interface·§3.5 The binary phase interface

Statement (paper)

For each edge of the complex network, write U,VU,V for its tail and head labels. Its phase-frame difference CVCU−1\mathcal C_V\mathcal C_U^{-1} is a product of one translation kernel aI+bXvaI+bX_v, or its inverse, per vector in an orthonormal basis of the edge residual. Here (Xvf)(x)=f(x+v)(X_vf)(x)=f(x+v). The sum of these basis lengths is sc=Wcm−2N+2Lc=1873807244636671267308000000<Wcm. s_{\rm c}=W_{\rm c}m-2N+2L_{\rm c} =1873807244636671267308000000<W_{\rm c}m. After source and sink diagonal corrections, and undoing the signed bank exchange, telescoping these phase frames gives C⊗mC^{\otimes m} on every role, including auxiliary roles. The same assertion holds on any positive number of columns by taking tensor products over columns.

In plain English

Each edge of the complex network has phase-frame difference CVCU−1\mathcal C_V\mathcal C_U^{-1} equal to a product of translation kernels aI+bXvaI+bX_v (or inverses), one per vector in an orthonormal basis of the edge residual, with total count sc=Wcm−2N+2Lc<Wcms_{\rm c}=W_{\rm c}m-2N+2L_{\rm c}<W_{\rm c}m. After diagonal corrections at source and sink and undoing the signed bank exchange, these frames telescope to C⊗mC^{\otimes m} on every role, auxiliary ones included, and this holds on any positive number of columns by tensoring.

Why it's here

The residual lemma supplies the orthonormal bases and the length count, so each edge's phase change splits into elementary kernels, and the positive deficit gives the bound sc<Wcms_{\rm c}<W_{\rm c}m. The signed-swap lemma then identifies the routes between roles, so the edge factors telescope to C⊗mC^{\otimes m} on every role. This is the interface that later layer constructions use, with cost measured by the number of kernels.
Classified routine:
This follows directly from the edge-residual lemma (which supplies the orthonormal bases and the length count s_c) and the signed-swap lemma (which identifies routes between roles). Factoring each phase-frame difference into translation kernels, one per basis vector, and telescoping along the network is bookkeeping on those earlier nodes.

Uses

lemEdge residuals and rank budgetlemForward invocation adds; three-stage swap with sign

Used by

propSimultaneous Hadamard layer in near-linear timeclaimRecurrence total cost and conversion to H

Inside the proof

claimPhase-frame difference factors into translation kernelsclaimKernel factorization extends to k columnsclaimRoute operators telescope to C^{⊗m}claimZ corrections repair the X_a to Y_a route

Sources

PaperOpen
Section
§3.5 The binary phase interface
Page
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Label
prop:complex-motif-interface
Line
sections/03-motifs.tex:568
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